A uniformly converging scheme for fractalconservation laws

Jerome Droniou and Espen R. Jakobsen

Abstract The fractal conservation law ∂tu + ∂x( f (u)) + (−∆)α/2u = 0 changescharacteristics as α → 2 from non-local and weakly diffusive to local and stronglydiffusive. In this paper we present a corrected finite difference quadrature methodfor (−∆)α/2 with α ∈ [0,2], combined with usual finite volume methods for the hy-perbolic term, that automatically adjusts to this change and is uniformly convergentwith respect to α ∈ [η ,2] for any η > 0. We provide numerical results which illus-trate this asymptotic-preserving property as well as the non-uniformity of previousfinite difference or finite volume type of methods.

1 Introduction

We consider the following fractional conservation law

∂tuα +∂x( f (uα))+Lα [uα ] = 0 , t > 0 , x ∈ R ,uα(0,x) = uini(x) , x ∈ R ,

(1)

where α ∈ [0,2], Lα = (−∆)α/2,

uini ∈ L∞(R)∩BV (R) and f : R→ R is locally Lipschitz-continuous. (2)

Such models appear for example in mathematical finance, gas detonation or semi-conductor growth [23, 26, 11, 1]. The fractional Laplacian Lα = (−∆)α/2 can be

Jerome DroniouSchool of Mathematical Sciences, Monash University, Victoria 3800, Australia, e-mail:jerome.droniou@monash.edu

Espen R. JakobsenDepartment of Mathematical Sciences, Norwegian University of Science and Technology, 7491Trondheim, Norway, e-mail: erj@math.ntnu.no

1

2 Jerome Droniou and Espen R. Jakobsen

defined e.g. as a Fourier multiplier, but for our purpose the following equivalent def-inition, valid for any ϕ ∈C∞

c (R) (set of smooth compactly supported functions), ismore useful:

L0[ϕ](x) = ϕ(x), α = 0,

Lα [ϕ](x) =−cα

∫R

ϕ(x+z)−ϕ(x)−ϕ ′(x)z1[−1,1](z)|z|1+α dz, α ∈ (0,2),

L2[ϕ](x) =−∆ϕ(x), α = 2,

(3)

where 1[−1,1] is the characteristic function of [−1,1], cα = (2π)α αΓ ( 1+α2 )

2π12 +α

Γ (1− α2 )

and

Γ is the Euler function [15].As α → 2, the operator Lα changes nature and properties. For α ∈ (0,2),

Lα is a non-local pseudo-differential operator of order < 2, and it has relativelyweak diffusive properties since the decay at infinity of the fundamental solution of∂tu+Lα [u] = 0 is polynomial. At α = 2, Lα =−∆ is a local operator with strongdiffusive properties and a fundamental solution with super-exponential decay. Whenα vary over [0,2], the qualitative behaviour of the solution uα of (1) also changes.In the case that α = 2, it is well-known that uα becomes instantly smooth for t > 0even when the initial data is discontinuous. On the contrary, for α = 0, the solu-tion may develop shocks and uniqueness of the solution requires additional entropyconditions and the corresponding notion of entropy solution [22]. The study of thefractional case α ∈ (0,2) dates back to [6], with some restrictions on α and f . Thefirst complete study in the case α > 1 for any locally Lipschitz f and bounded initialdata uini can be found in [14]. Here it is proved that the solution becomes instantlysmooth even if uini is only bounded (see also [15]). If α < 1, then the solution candevelop shocks [4] and the weak solution need not be unique [3]. The notion ofentropy solution of [2] is therefore required to obtain a well-posed formulation.

There exists a vast literature on the numerical approximation of scalar conser-vation laws (i.e. (1) without Lα ), see e.g. [17, 18, 19] and references therein. Thestudy of numerical methods for fractal conservation laws is much more recent witha corresponding less extensive literature. Probabilistic methods have been studied in[21, 24], but must be applied to the equation satisfied by ∂xuα in order to avoid noisyresults, and recovering from this a numerical approximation of uα may be challeng-ing in dimension greater than 1. Deterministic methods for (1) like finite difference,volume, and element methods (discontinuous Galerkin) are given in [13, 8, 10],while a high order spectral vanishing viscosity method is introduced in [9]. The lat-ter method and its analysis is very different from the former three methods, withconvergence and (non-optimal) error estimates that are independent of α ∈ (0,2).As opposed to the spectral method, the other methods are monotone or have loworder monotone variants.

Surprisingly, for all the non-spectral monotone methods the convergence dete-riorates as α → 2, and the schemes themselves are not even defined in the limitα = 2. The purpose of this paper is to present an asymptotic-preserving monotonescheme for (1) defined for any α ∈ [0,2], i.e. a scheme that provides a monotone

A uniformly converging scheme for fractal conservation laws 3

approximation of uα which is uniform with respect to α ∈ [0,2]. In particular, ourscheme naturally adapts to the change of behaviour of Lα as α → 2 and α → 0and its convergence properties do not deteriorate in these extreme cases. The ideabehind our scheme is to add a correction term in the form of a suitably chosen van-ishing local viscosity term. Similar ideas have been used for other equations before,see e.g. [12] for linear equations and [20] for fully nonlinear equations. A stochasticinterpretation can be found in [5].

This paper is organised as follows. The numerical method is presented in Section2, and its asymptotic-preserving characteristics are discussed. Due to lack of spaceand the technical nature of the proofs, we skip them and refer instead to [16]. InSections 3 and 4, we define precisely what asymptotic preserving means and the wegive a couple numerical simulations to illustrate this property of the method.

2 The scheme

The new scheme is based on monotone convervative finite difference approxima-tions of the local terms combined with quadrature, truncation of 1

|z|1+α near the sin-gularity, and a second order correction term (vanishing viscosity) for the non-localterm. Except for the correction term, the scheme is similar to the schemes of [13, 8]and of [10] with P0-elements. It is monotone, conservative, and converges in L1

locuniformly in α ∈ [η ,2] for all η > 0.

For given space and time steps δ x,δ t > 0, we introduce the grid tn := nδ t andxi := iδ x+ δ x

2 for n ∈N0 and i ∈ Z. We identify sequences (ϕi)i∈Z of numbers withpiecewise constant functions ϕδ x : R→ R equal to ϕi on [iδ x,(i+ 1)δ x) for alli ∈ Z. Similarly, (ϕn

i )n≥0 , i∈Z is identified with ϕδ x,δ t : [0,∞)×R→ R equal to ϕni

on [nδ t,(n+ 1)δ t)× [iδ x,(i+ 1)δ x) for all n ≥ 0 and i ∈ Z. The discretisation of(1) can then we written as: find uα,δ x,δ t = (un

i )n≥0 , i∈Z such that

u0i =

1δ x

∫[iδ x,(i+1)δ x)

u0(x)dx for all i ∈ Z, (4)

un+1i −un

iδ t

+Fδ x(un)i +Lα,δ x[u

n+1]i = 0 for all n≥ 0 and all i ∈ Z, (5)

where Fδ x is any monotone consistent and consevative discretization of ∂x( f (u))(see e.g [17, 18, 19]), and Lα,δ x is a monotone discretisation of Lα to be defined.Note that the scheme has explicit convection and implicit diffusion terms.

The first and simplest idea to obtain a monotone discretization of Lα for α ∈(0,2) is to discretize the integral in (3) using a simple (weighted) midpoint typequadrature rule, see e.g. [13, 10, 8]. For ϕ ∈C∞

c (R) and letting ϕl = ϕ(xl) if l ∈ Z,this leads to

Lα [ϕ](xi)≈ Lα,δ x[ϕ]i :=− ∑j∈Z\{0}

(ϕi+ j−ϕi

)∫( jδ x− δ x

2 , jδ x+ δ x2 )

cα

|z|1+αdz. (6)

4 Jerome Droniou and Espen R. Jakobsen

However, as α → 2 we have cα → 0 and therefore Lα,δ x → 0 for fixed δ x. In thelimit α → 2 the scheme then converges to

un+1i −un

iδ t

+Fδ x(un)i = 0 for all n≥ 0 and all i ∈ Z,

which is a discretisation of ∂tu+ ∂x( f (u)) = 0 and not ∂tu+ ∂x( f (u))−∆u = 0.Hence the limits α → 2 and δ x → 0 do not commute and the scheme is notasymptotic-preserving.

Note that Lα,δ x vanishes in the limit because the measure cα dz|z|1+α concentrates

around 0 as α → 2, while in the above midpoint rule the integral in (3) over(− δ x

2 , δ x2 ) will always be zero by symmetry. We therefore need to replace the mid-

point rule on this interval by a more accurate rule based on the second order in-terpolation polynomial Pi of ϕ around the node xi. We find that this polynomialsatisfies Pi(xi + z)−Pi(xi)−P′i (xi)z = 1

2δ x2

(z2ϕi−1−2z2ϕi + z2ϕi+1

)and the new

discretization therefore becomes

Lα,δ x[ϕ]i :=−cα

∫ δ x2

− δ x2

P(xi + z)−P(xi)−P′(xi)z|z|1+α

dz+ Lα,δ x[ϕ]i

=ϕi+1−2ϕi +ϕi−1

δ x2

∫(− δ x

2 , δ x2 )

cα |z|1−α

2dz+ Lα,δ x[ϕ]i.

We can check that the new approximation has the following truncation error [16]:

|Lα [ϕ](xi)− Lα,δ x[ϕ]i|

≤ C(‖ϕ(4)‖L∞δ x4−α +‖ϕ ′′‖L∞cα(

1α+ 1|1−α| )δ xmin(1,2−α)+‖ϕ ′‖L∞δ x

),

which is O(δ x)+ oα(1) as α → 2 and therefore does not deteriorate in this limit.Note that if α = 1, then 1

|1−α|δ xmin(1,2−α) must be replaced with δ x| ln(δ x)|.

In order to obtain an approximation which uses only a finite number of discretevalues, we also truncate the sum in (6) as in [13] at some index Jδ x > 0 (which maydepend upon α) where Jδ xδ x→∞ as δ x→ 0. The final approximate operator Lα,δ xis therefore

Lα,δ x[ϕ]i =− ∑0<| j|≤Jδ x

W jα,δ x(ϕi+ j−ϕi)−W Jδ x+1

α,δ x

(ϕi−Jδ x−1−ϕi

)−W Jδ x+1

α,δ x

(ϕi+Jδ x+1−ϕi

)−W 0

α,δ xϕi+1−2ϕi +ϕi−1

δ x2 , (7)

with weights

A uniformly converging scheme for fractal conservation laws 5

W 0α,δ x =

∫(− δ x

2 , δ x2 )

cα |z|1−α

2dz ,

W jα,δ x =

∫( jδ x− δ x

2 , jδ x+ δ x2 )

cα

|z|1+αdz for 0 < | j| ≤ Jδ x,

W Jδ x+1α,δ x =

∫z>Jδ xδ x+ δ x

2

cα

|z|1+αdz =

∫z<−Jδ xδ x− δ x

2

cα

|z|1+αdz.

(8)

The last term in (7) contains the classical discretization of ϕ ′′(xi) and is the newcorrection term compared with the discretisations of [13, 10, 8]. Discretisation (7)–(8) fits in the generic framework of [13] from which we can conclude:

Theorem 1 ([16]). Under a standard CFL condition for the convection term,

1. There is a unique solution uα,δ x,δ t of the scheme defined by (4), (5), (7) and (8),satisfying ‖uα,δ x,δ t‖L∞ ≤ ‖uini‖L∞ and |uα,δ x,δ t(t, ·)|BV ≤ |uini|BV for all t > 0.

2. For fixed α , uα,δ x,δ t converges in L1loc([0,∞)×R) as (δ x,δ t)→ 0 to the unique

entropy solution uα of (1).

Remark 1. We set L2,δ x[ϕ]i = −(ϕi+1−2ϕi +ϕi−1)/δ x2 and L0,δ x[ϕ]i = ϕi. Thisconsists in fixing δ x and sending α → 2 or α → 0 in (7). Taking the limits in thescheme (5), we obtain the classical implicit scheme for the (1) with α = 2 or α = 0.

3 The asymptotic-preserving property

The scheme is asymptotic-preserving if its solution uα,δ x,δ t satisfies the followinguniform approximation result away from α = 0 (see [16] for the case α = 0):

∀η > 0 , supα∈[η ,2]

dL1loc([0,∞)×R)(uα,δ x,δ t ,uα)→ 0 as (δ x,δ t)→ 0 (9)

where dL1loc([0,∞)×R)(u,v) = ∑

∞n=1 2−n min(1, ||u− v||L1([0,n)×(−n,n))) is the usual dis-

tance defining the topology of L1loc([0,∞)×R). Here and elsewhere, the convergence

(δ x,δ t)→ 0 is always taken under a standard CFL condition depending on the def-inition of the convective flux F in (5) (see e.g. [13, 10, 8]). This formulation ofthe asymptotic-preserving property is very general and does not require an expliciterror estimate independent on α . Such an estimate seems particularly challengingto obtain in the absence of regularity of the solution as t→ 0.

Theorem 2 ([16]). Under a standard CFL for the convection part, the numericalscheme defined by (4) (5), (7) and (8) is asymptotic-preserving.

Next we want to illustrate this property numerically. As it is formulated now, thiswould require us to have access to the exact solution uα , which is not the case. Weovercome this difficulty by using instead the following equivalent reformulation of(9) (see [16]), which can be checked by computing approximate solutions only:

6 Jerome Droniou and Espen R. Jakobsen

∀α0 ∈ (0,2] , for any sequence (δ xk,δ tk)k∈N converging to 0:supk≥1

dL1loc([0,∞)×R)(uα,δ xk,δ tk

,uα0,δ xk,δ tk)→ 0 as α → α0. (10)

Remark 2. The matrix of Lα,δ x defined by (7) is a semi-definite Toepliz matrix as in[13, 10, 8]. Implementation of the scheme thus takes advantage of super-fast mul-tiplication and inversion algorithms for these matrices [7, 25]. Computing severalapproximate solutions, as required in (10), is therefore not very expensive.

4 Numerical results

In all these tests, we take the Burgers flux f (u) = u2

2 and Fδ x given by a MUSCLmethod. The final time is T = 1 and the spatial computational domain is [−1,1]. Weuse the same truncation parameters (in particular Jδ x) as in [13, Section 4.1.2].

For each test, we choose the discretisation steps (δ xk,δ tk) = ( 12k×50 ,

12k×100 )

for k = 1, . . . ,4, which all satisfy the CFL for (5). We also select four values(αm)m=1,...,4 = (1.8,1.9,1.99,1.999) which are near α0 = 2, the difficult case in as-sessing the uniformity of the convergence in (10) and the reason why we introducedthe correction term in (7). We then indicate, for m = 1, . . . ,4, the value of

Em = maxk=1,...,4

supt∈[0,1]

||uαm,δ xk,δ tk(·, t)−uα0,δ xk,δ tk

(·, t)||L1([−1,1]),

that is the maximum over k = 1, . . . ,4 of the L∞(L1) norm of uαm,δ xk,δ tk−uα0,δ xk,δ tk

on the computational domain. This is a stronger norm that the L1(L1) norm used in(10). Hence, Em approaching 0 as m increases is an even better indication that thescheme is asymptotic-preserving.

Test 1 (rarefaction): we select a Riemann initial condition, uini = −1 if x < 0and uini = 1 if x > 0. In this case both convection and diffusion work to smooth outthe intial data. Table 1 shows the values of (Em)m=1,...,4 for both the uncorrectedscheme from [13] based on (6) and our corrected scheme based on (7).

Table 1 Comparison be-tween the uncorrected schemeof [13] and our correctedscheme, uini =−1 on (−∞,0),uini = 1 on (0,∞).

E1 E2 E3 E4Uncorrected scheme 1.8E-1 3E-1 8.8E-1 9.1E-1Corrected scheme 5.1E-2 2.2E-2 1.7E-4 1.7E-5

Test 2 (smooth shock): the initial condition is uini(x) = 1 if x < 0 and uini(x) =−1 if x > 0. Here the hyperbolic and non-local terms in (1) compete to maintainor diffuse the initial shock. Since αm is near 2 however, any solution is instantlysmooth, but has much larger gradients near x = 0 than the solution in Test 1 (Table2).

A uniformly converging scheme for fractal conservation laws 7

Table 2 Comparison be-tween the uncorrected schemeof [13] and our correctedscheme, uini = 1 on (−∞,0),uini =−1 on (0,∞).

E1 E2 E3 E4Uncorrected scheme 2.1E-1 3.9E-1 1.3 1.3Corrected scheme 5.3E-2 2.3E-2 3.2E-4 4.2E-5

Both tests confirm that the scheme defined by (4), (5), (7) and (8) is asymptotic-preserving. They also confirm that, without the order 2 correction in (7), the schemedeteriorates as α → 2 and does not provide a correct numerical solution at anyreasonable resolution. This is also illustrated in Figure 1, where we plot the solutionsof both schemes for α = 1.99 for the initial condition of Test 2 and (δ x,δ t) =( 1

24×50 ,1

24×100 ). Even at this very high resolution, the uncorrected scheme providesan incorrect approximate solution which, as expected, is closer to the solution of∂tu+∂x( f (u)) = 0 than to the solution of (1).

−1 −0.5 0.5 1

−1

−0.5

0.5

1

Fig. 1 Approximate solutionsprovided at T = 1 by thecorrected (continuous) anduncorrected (dashed) schemesfor (1) with α = 1.99. Thedotted line is both the initialcondition and the solution to∂t u+∂x( f (u)) = 0.

5 Conclusion

We have presented a monotone numerical method for fractional conservation lawswhich is asymptotic-preserving with respect to the fractional power of the Lapla-cian. The scheme automatically adjusts to the change of nature of the equation asthe power of the Laplacian goes to 1 (i.e. α→ 2 in (1)) and therefore provides accu-rate approximate solutions for any power of the fractional Laplacian. We have givennumerical results to illustrate the asymptotic-preserving property of our method, aswell as the necessity of modifying previously studied monotone methods to obtainthis property.

The complete theoretical study of such monotone asymptotic-preserving schemeswill be presented in the forthcomming paper [16]. Here a general class of fractionaldegenerate parabolic equations are considered that include (1) as a special case.

8 Jerome Droniou and Espen R. Jakobsen

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